Prove that the series SUM (-1)^n n/p_n converges where p_n are primes

  • Thread starter Dragonfall
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  • #1
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Homework Statement



Prove that [tex]\sum(-1)^n\frac{n}{p_n}[/tex] converges, where [tex]p_n[/tex] is the nth prime.

Homework Equations



The sequence [tex]\frac{n}{p_n}[/tex] is definately not monotone if there exists infinitely many twin primes, since [tex]2n-p_n<0[/tex] for sufficiently large n, so alternating series test is out. Are there any other ways of showing this converges?
 

Answers and Replies

  • #2
StatusX
Homework Helper
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Can you use the prime number theorem? This says that:

[tex] \lim_{n \rightarrow \infty} \frac{p_n}{n \ln n} = 1 [/tex]
 
  • #3
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I can't use it for the series. I can only establish that n/p_n -> 0, which is insufficient for the series. I can't even prove that n/p_n is NOT monotone for large n, unless I assume the twin prime conjecture, for example.
 

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